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Exponential decay in the frequency of analytic ranks of automorphic \(L\)-functions. (English) Zbl 1166.11326
From the text: This note should be seen as an addendum to the work of E. Kowalski and the second author [Duke Math. J. 100, 503–542 (1999; Zbl 1161.11359)] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of \(L\)-functions. This problem is illustrated in the particular case of \(L\)-functions of weight-2 primitive modular forms of prime level.
Recall the notation from that paper: For q a prime number, let \(S_2(q)^*\) be the set of primitive forms of weight 2 and level \(q\), normalized so that their first Fourier coefficient is 1; for \(f (z) =\sum_{n\geq 1} \lambda_f(n)n^{1/2}e(nz) \in S_2(q)^*\), \(\lambda_f (1) = 1\), let \(L(f, s) :=\sum_{n\geq 1}\lambda_f (n)n^{-s}\), the associated (normalized) \(L\)-function; it admits analytic continuation to \(\mathbb C\) with a functional equation relating \(L(f, s)\) to \(L(f,1-s),\) and we call \(r_f := \text{ord}_{s=1/2} L(f, s)\) the analytic rank of \(f\) .
In [Kowalski-Michel (loc. cit.)] the following was proved: There exists an absolute constant \(C_1 \geq 0\) such that, for all \(q\) prime, \(\sum_{f \in S_2(q)^*} r_f \leq C_1\,| S_2(q)^*|.\)
After that, much progress has been made concerning this question. In particular, in [E. Kowalski, P. Michel and J. M. VanderKam, J. Reine Angew. Math. 526, 1–34 (2000; Zbl 1020.11033)], a sharp explicit value was given for the constant \(C_1\) (\(C_1 = 1.1891\) for \(q\) large enough). In the course of the proof, a uniform bound for the square of the ranks was obtained: \(\sum_{f \in S_2(q)^*}r_f ^2 \leq C_2\,| S_2(q)^*|.\)
However, the latter improvement used only a slight variant of the methods of Kowalski-Michel (loc. cit.). In fact, it is possible to pursue this idea further and it turns out that much more is true; this is the subject of the present note.
Consider a finite probability space \((\Omega,\mu)\), where \(\mu(\omega) > 0\) for every \(\omega\in \Omega\). For each \(\omega\in \Omega\), suppose given a function \(h_\omega(s)\) which is holomorphic in the half plane \(\text{Re}(s) \geq 0\). Moreover assume the following hypothesis on the variance of the function \(h_\omega(s)-1\):
Hypothesis: For some \(B,C > 0\), \(M > 2\), we have the bound
\[ \sum_{\omega\in \Omega}|h_\omega(\sigma + it) - 1|^2\mu(\omega) \leq C(1 + |t|)^BM^{-\sigma} \] uniformly for \(\sigma\geq (2 \log M)^{-1}\).
Note that in view of this hypothesis and the fact that \(\mu(\omega) > 0\), we have
\[ h_{\omega}(s) = 1 + O_{\omega}((1 + |t|)^{B/2}M^{-\sigma/2}) \tag{(*)} \]
for each \(h_{\omega}\). Thus \(h_{\omega}(s)\) is nonvanishing for sufficiently large \(\sigma\). For any \(\alpha\geq 0\) and \(t_1 < t_2 \in \mathbb R\), we may therefore define \(N(\omega, \alpha, t_1, t_2)\) to be the number of zeros \(\rho\) of \(h_{\omega}(s)\) such that \(\text{Re}(\rho)\geq \alpha\), \(t_1\leq \text{Im}(\rho)\leq t_2\). Clearly \(N(\omega, \alpha, t_1, t_2)\) is finite. Our general result gives an upper bound for the \(2k\)-th power of \(N(\omega, \alpha, t_1, t_2)\) on average:
Theorem: With the above notations, assume that the above Hypothesis is satisfied. Then for all \(k\geq 1\), for all \(\alpha\geq (\log M)^{-1}\), and all \(t_1 < t_2,\) we have
\[ \sum_{\omega\in \Omega} N(\omega, \alpha, t_1, t_2)^{2k}\mu(\omega)\ll C(k!)^2\left(48 \frac{k}{\alpha \log M}\right)^{2k} \left(1+|t|+\frac{16k}{\log M}\right)^BM^{-\alpha/2}(1+(t_2-t_1) \log M) \]
where we have set \(|t| := \max(|t_1|, |t_2|)\). The constant involved in the Vinogradov symbol is absolute.

MSC:
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
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References:
[1] É. Fouvry, “Sur le comportement en moyenne du rang des courbes \(y^ 2=x^ 3+k\)” in Séminaire de Théorie des Nombres (Paris, 1990–91.) , Progr. Math. 108 , Birkhäuser, Boston, 1993, 61–84. · Zbl 0814.11034
[2] D. R. Heath-Brown, The size of Selmer groups for the congruent number problem, Invent. Math. 111 (1993), 171–195. · Zbl 0808.11041 · doi:10.1007/BF01231285 · eudml:144075
[3] –. –. –. –., The size of Selmer groups for the congruent number problem, II, Invent. Math. 118 (1994), 331–370. · Zbl 0815.11032 · doi:10.1007/BF01231536 · eudml:144239
[4] E. Kowalski and P. Michel, The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions , Duke Math. J. 100 (1999), 503–542. · Zbl 1161.11359 · doi:10.1215/S0012-7094-99-10017-2
[5] ——–, An explicit upper bound for the rank of \(J_{0}(q)\) , to appear in Israel J. Math.
[6] E. Kowalski, P. Michel, and J. M. VanderKam, Non-vanishing of high derivatives of automorphic \(L\)-functions at the center of the critical strip , to appear in J. Reine Angew. Math. · Zbl 1020.11033 · doi:10.1515/crll.2000.074
[7] M. Ram Murty, “The analytic rank of \(J_0(N)\)(\(\mathbb{Q}\))” in Number Theory (Halifax, N.S., 1994), CMS Conf. Proc. 15 , Amer. Math. Soc., Providence, 1995, 263–277. · Zbl 0851.11036
[8] A. Selberg, Contributions to the theory of Dirichlet’s \(L\)-functions , Skr. Norske Vid.-Akad. Oslo I Mat.-Nat. Kl. 1946 , no. 3. · Zbl 0061.08404
[9] K. Soundararajan, Non-vanishing of quadratic Dirichlet \(L\)-functions at \(s=1/2\) , preprint, 1999. JSTOR: · Zbl 0964.11034 · doi:10.2307/2661390 · www.math.princeton.edu · eudml:121051
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